File:Stability region for BDF1.svg
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Original file (SVG file, nominally 720 × 540 pixels, file size: 88 KB)
Summary
| Description |
English: Region of absolute stability for the backward Euler = BDF1 method. See below for Python source. Compare with page 350 of Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0521007941. |
| Date | |
| Source | Own work |
| Author | Jitse Niesen |
| SVG development |  This plot was created with Matplotlib. |
| Source code | Python codeimport numpy
from matplotlib import pyplot
BFDcoeffs = { 1: {'alpha': [1, -1], 'beta': 1},
2: { 'alpha': [3, -4, 1], 'beta': 2 },
3: { 'alpha': [11, -18, 9, -2], 'beta': 6 },
4: { 'alpha': [25, -48, 36, -16, 3], 'beta': 12 },
5: { 'alpha': [137, -300, 300, -200, 75, -12], 'beta': 60 },
6: { 'alpha': [147, -360, 450, -400, 225, -72], 'beta': 60 } }
plotWindow = { 1: { 'realPart': [-2, 3], 'imagPart': [-2, 2] },
2: { 'realPart': [-2, 5], 'imagPart': [-3, 3] },
3: { 'realPart': [-4, 8], 'imagPart': [-5, 5] },
4: { 'realPart': [-4, 14], 'imagPart': [-8, 8] },
5: { 'realPart': [-10, 25], 'imagPart': [-15, 15] },
6: { 'realPart': [-20, 40], 'imagPart': [-30, 30] } }
# Returns > 1 if argument is not in region of absolute stability
def stabilityFunction(hTimesLambda, s):
stabPolyCoeffs = list(BFDcoeffs[s]['alpha'])
stabPolyCoeffs[0] -= hTimesLambda * BFDcoeffs[s]['beta']
return max(abs(numpy.roots(stabPolyCoeffs)))
# Main program
for s in range(1,7):
x = numpy.linspace(*plotWindow[s]['realPart'], num=400)
y = numpy.linspace(*plotWindow[s]['imagPart'], num=400)
[X,Y] = numpy.meshgrid(x,y)
Z = numpy.zeros(X.shape)
for m in range(X.shape[0]):
for n in range(X.shape[1]):
Z[m,n] = stabilityFunction(X[m,n] + 1j * Y[m,n], s)
pyplot.contour(X, Y, Z, [1], colors='k')
pyplot.contourf(X, Y, Z, [0,1], colors=[[1, 0.5, 0.8]])
pyplot.plot(plotWindow[s]['realPart'], [0, 0], 'k--')
pyplot.plot([0, 0], plotWindow[s]['imagPart'], 'k--')
pyplot.gca().tick_params(labelsize = 20)
pyplot.savefig('Stability_region_for_BDF%d.svg' % s)
pyplot.clf()
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Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 17:37, 20 March 2012 | | 720 × 540 (88 KB) | wikimediacommons>Jitse Niesen |
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